assessment of goldmann tonometry using numerical modelling
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Evaluation of Goldmann Applanation Tonometry Using a Nonlinear FiniteElement Ocular Model
AHMED ELSHEIKH ,1
DEFU WANG,1
AACHAL KOTECHA,2,3
MICHAEL BROWN,1
and DAVID GARWAY-HEATH2,3
1Division of Civil Engineering, Faculty of Engineering, University of Dundee, Dundee, DD1 4HN, UK; 2Glaucoma ResearchUnit, Moorfields Eye Hospital, London, EC1V 2PD, UK; and 3Department of Optometry and Visual Science, City University,
London, UK
(Received 22 February 2006; accepted 29 August 2006; published online: 28 September 2006)
AbstractGoldmann applanation tonometry (GAT) is theinternationally accepted standard for intra-ocular pressure(IOP) measurement, which is important for the diagnosis ofglaucoma. The technique does not consider the effect of thenatural variation in the corneal thickness, curvature and
material properties. As these parameters affect the structuralresistance of the cornea, their variation is expected to lead toinaccuracies in IOP determination. Numerical Analysis basedon the finite element method has been used to simulate theloading conditions experienced in GAT and hence assess theeffect of variation in corneal parameters on GAT IOPmeasurements. The analysis is highly nonlinear and considersthe hyper-elastic J-shaped stressstrain properties of cornealtissue observed in laboratory tests. The results reveal a clearassociation between both the corneal thickness and materialproperties, and the measured IOP. Corneal curvature has aconsiderably lower effect. Similar trends have been foundfrom analysis of clinical data involving 532 patients referredto the Glaucoma Unit at Moorfields Hospital, and fromearlier mathematical analyses. Nonlinear modelling is shownto trace the behaviour of the cornea under both IOP andtonometric pressure, and to be able to provide additional,and potentially useful, information on the distribution ofstress, strain, contact pressure and gap closure.
KeywordsTonometry, Intra-ocular pressure, Cornea,
Numerical modelling.
ABBREVIATIONS
CCT central corneal thickness
GAT Goldmann applanation tonometry
IOP intra-ocular pressureIOPG intra-ocular pressure as measured by GAT
IOPT true intra-ocular pressure
PCT peripheral corneal thickness
INTRODUCTION
Intra-ocular pressure (IOP) measurement in tonom-
etry is important for the diagnosis and management of a
number of conditions, most notably glaucoma, thesecond most common cause of irreversible blindness
in the world. Goldmann applanation tonometry (GAT)
developed in the mid-1950s, is still the internationally
accepted standard for IOP determination. It makes a
pseudo-static measurement of the force required to
flatten a fixed area of the central cornea and uses
this force to estimate the value of IOP. The natural
variation in the central corneal thickness (CCT),
corneal curvature and material properties, which have a
direct bearing on the corneal structural resistance,
can affect the accuracy of IOP measurement, and
since IOP is a major risk factor for glaucoma and
forms part of the classification of ocular hypertension
and normal tension glaucoma,13 errors in IOP
measurement can lead to the misdiagnosis of these
conditions.
The possible impact of corneal thickness on IOP
measurement using GAT was identified and discussed
briefly by Goldmann and Schmidt.16 Later studies by
Ehlers and co-workers9 drew attention to the effect of
CCT on IOP measurement, and the interest in this
effect grew further with the advent of refractive surgery
procedures involving an iatrogenic thinning of the
cornea, see Refs. [5 ,7,18,29,32]. The overall conclusion
that can be drawn from these studies is that IOPmeasurements using tonometry are affected by differ-
ences in CCT. While the vast majority of the studies
were based on statistical analyses of clinical data,
mathematical analysis was successfully used in a
number of studies including those by Orssengo and
Pye29 and by Liu and Roberts25 and produced results
with a similar trend. All found that high CCT led to
IOP overestimations while low CCT led to IOP
underestimations. However, there is no agreement yet
Address correspondence to Ahmed Elsheikh CEng MICE PhD,
Division of Civil Engineering, Faculty of Engineering, University of
Dundee, Dundee, DD1 4HN, UK. Electronic mail: a.i.h.elsheikh@
dundee.ac.uk
Annals of Biomedical Engineering, Vol. 34, No. 10, October 2006 ( 2006) pp. 16281640
DOI: 10.1007/s10439-006-9191-8
0090-6964/06/1000-1628/0 2006 Biomedical Engineering Society
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on the pressure correction factors that should be used
to consider CCT variations.
Other sources of tonometry errors include the cor-
neal curvature as has been reported by Gunvant
et al.,18 Liu and Roberts25 and Kanngiesser et al.23
Furthermore, wound healing following surgical pro-
cedures is also believed to lead to changes in the bio-
mechanical properties of corneal tissue. These changes,
although not yet quantified, are expected to influence
the structural resistance of the cornea and might
therefore affect the accuracy of IOP measurements in
tonometry.25
The aim of this study is to use numerical modelling
based on nonlinear finite element analysis to create a
representative model of the GAT procedure. Numeri-
cal modelling is adopted as it has the potential to
represent real life conditions without having to adopt
the simplifications necessary with mathematical closed
form solutions. The development of the numerical
model has undergone a number of stages to optimise
its construction and improve its accuracy. The com-plexity of the ocular structure, both at the microscopic
and macroscopic levels, makes it essential to distin-
guish between the parameters that have a considerable
effect on behaviour (and which should therefore be
incorporated in model construction), and the param-
eters which can be ignored for their negligible effect.
The construction of the model was followed by a val-
idation process against experimental tests before it was
used in a parametric study in which GAT procedure
was simulated with different values of corneal thick-
ness, curvature and material properties. The results are
compared against the outcome of statistical analysis ofnew clinical data and the results of earlier mathemat-
ical analysis.
MODEL CONSTRUCTION
Nonlinear finite element modelling was used in this
research to enable a detailed representation of biome-
chanical behaviour and to provide a systematic
approach to determine the impact of variation in cor-
neal parameters on tonometry. The complexity of the
structure and form of the cornea at both the micro-
scopic and macroscopic levels presented a particular
challenge during the development of the numerical
models. On one hand, there was a desire to simulate
the real structure of the cornea in order to improve
accuracy, but on the other there was a practical
requirement to simplify the models and keep them at a
reasonable level of complexity to reduce computa-
tional cost. In order to strike the best balance between
computational cost and accuracy, a study10 was con-
ducted to identify the effect of individual parameters
on the models behaviour. The parameters that were
found to have a small or a negligible effect (with an
effect on results below 1%) were not considered in the
final construction of the model.
The parameters considered in the study10 were the
thickness variation between the limbus and the corneal
centre, representation of the boundary conditions
along the edge of the cornea, the material properties
and the corneal topography. The density of the finite
element mesh and the cornea discretisation method
were also considered. According to the findings of this
study, an optimum model construction involves the
following details:
The discretisation method shown in Fig. 1 is used toensure all element internal angles are kept within
practical limits (20 and 80). The model follows
the structural form of diamatic skeletal domes built
in structural engineering applications.27
Solid six-noded elements are used with six degrees of
freedom per node (u, v, w, hx, hy, hz). These elementshave been found to closely simulate the behaviour of
experimental test specimens while enabling a good
representation of corneal variable thickness.
The model has a number of element layers to enabletracing the stress and strain distributions across the
corneal thickness. The number of layers is at least
two under uniform pressures (e.g. IOP) and six
under concentrated effects such as point loads and
tonometry pressures. This feature is also important
for future development of the numerical model as it
FIGURE 1. Discretisation method adopted in construction ofcorneal modelmodel shown has 6 segments, 7 rings and294 elements per layer.
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allows the use of different material properties for the
epithelium, endothelium and stroma when they
become available.
The minimum number of elements in each layer is294, arranged in six segments and seven rings.
Although the model predictions with this number of
elements are close to experimental observations,
more elements, and hence a finer mesh, would be
needed if smooth contour distributions are to be
obtained.
The thickness variation between a minimum at thecentre to a maximum along the limbus is modelled
to approximate natural topography.
The corneal model is provided with edge rollersupports oriented at 23 to the limbal plane. This
choice of boundary conditions and orientation angle
creates working conditions similar to those created
by the actual connection to the sclera, see Fig. 2. In
arriving at this angle, a whole eye model was
compared to a cornea-only model provided with
edge roller supports. The angle or orientation of theroller supports was varied until the behaviour
predictions of the two models under both distributed
pressure and concentrated load were in close agree-
ment (less than 2% on average). Details of this study
can be found in Refs. [1,10].
The corneo-scleral intersection is oval (elliptical)with the temporalnasal diameter (DTN) 10% larger
than the inferiorsuperior diameter (DIS). At the
same time, constant curvature is assumed along the
corneas two main meridian lines. This choice of
topography, which represents an approximation of
the actual aspheric form of the cornea, was neces-
sary because of (1) the lack of available corneal
topography maps and (2) the need to adopt an easy
to define and modify topography in the numerical
models.
The nonlinear material properties of corneal tissueobserved in laboratory tests1 are incorporated in the
model. The stressstrain relationship observed dem-
onstrates clear hyper-elastic behaviour with an
initial low stiffness phase followed by another with
much higher stiffness. The effect of using a simpli-
fied linear-elastic material model in Goldmann
tonometry modelling is illustrated in this paper.
The nonlinear material stressstrain property is
incorporated using a hyper-elastic material model
based on Ogdens strain energy function28:
UXN
i1
2ui
a2i
kai1 kai2 k
ai3 3
XN
i1
1
DiJel 12i
1
where Uis the strain energy per unit volume, ki are the
principal stretches, N is a material parameter defining
Angle
(a)
0
0.005
0.01
0.015
0.02
0.025
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Central corneal rise (mm)
Intra-ocularpr
essure(N/mm
2) Cornea model - 15 roller supports
Cornea model - 20 roller supports
Cornea model - 23 roller supports
Cornea model - 30 roller supports
Cornea model - 40 roller supports
Whole eye model
(b)
FIGURE 2. Corneal model with roller edge supports to simulate actual connection with the sclera. (a) Angle h found from com-parisons with a full ocular model to be 23 (Ref. [10]), (b) comparison of behaviour predictions between a whole eye model and acornea only model under IOP and with different angles h.
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the order of the equation, ui, ai, and Di are tempera-
ture-dependent material parameters and Jel is the
elastic volume ratio.
The Abaqus software package20 is used in this work.
The analyses consider both geometric nonlinearities due
to change of joint coordinates and material nonlinear-
ities. In tracing the nonlinear behaviour the Riks arc
method31 is adopted. In this method, load increments
vary according to the current stage of overall behaviour
and are controlled automatically such that a solution is
obtained even after the point of mechanical failure.
CORNEAL THICKNESS
Several recent studies measured corneal thickness in
recognition of its likely effect on tonometry measure-
ments. Most studies concentrated on the central
corneal thickness and used analysis of clinical obser-
vations to give its average value and range of variation.
The average values of CCT vary slightly betweenstudies, for instance: 0.548 mm is reported by Cho and
Cheung4 and 0.545 mm for the right eye and 0.547 mm
for the left eye by Lam and Chan.24 The natural var-
iation of CCT in the population was also estimated in
earlier studies including Feltgen et al.12 who reported
values between 0.448 and 0.713 mm. In the present
numerical study, a wider range of variation in CCT is
considered; between 0.320 and 0.720 mm to go beyond
the natural range.
The relationship between the CCT and the periph-
eral corneal thickness (PCT) is not yet established.
While it is known that PCT is always larger than CCT,
it is not clear whether PCT changes with CCT, and if
so by what degree. In this study, it is assumed that PCT
is always 0.150 mm larger than CCT. This is compat-
ible with the corneal dimensions of the Gullstrands
No. 1 schematic eye,1 in which CCT = 0.520 mm and
PCT = 0.670 mm.
CORNEAL CURVATURE
Corneal curvature variation has been considered as
another possible cause of error in tonometry mea-
surements.6 The radius of anterior curvature, R, theterm used to describe corneal curvature, is found by
Doughty and Zaman6 to change between 7.53 and
8.09 mm for children, 7.58 and 8.14 mm for adults,
and 7.36 and 7.86 mm for elderly adults. Similar val-
ues, between 7.60 and 8.14 mm, are reported by
Dubbelman et al.8 A wider range of variation in R
between 7.20 and 8.40 mm is considered in this study
to determine the effect of corneal curvature on IOP
measurements.
CORNEAL MATERIAL PROPERTIES
The material properties of corneal tissue clearly
have an effect on the structural resistance of the cor-
nea, and hence the accuracy of IOP measurement.
Despite this logical association, the material properties
have rarely been considered an important parameter in
earlier attempts to improve the accuracy of tonometry.
A recent exception is the work published by Liu andRoberts25 which found the material properties to have
a profound effect on GAT IOP.
There is a wide variation between the material
properties reported in earlier publications.33 The vari-
ation seems to be related to three main factors, the
pressure or stress at which Youngs modulus (E) is
determined, the test method and the test strain
rate. Hoeltzel et al.,21 for instance, reports two values
of E, 0.34 N/mm2 under IOP = 10 mmHg (0.0013 N/
mm2) and 4.1 N/mm2 under IOP = 400 mmHg
(0.053 N/mm2). Kampeier et al.22 also reports a lower
E = 0.4 N/mm2
at strain = 0.02 and a higher E =3.00 N/mm2 at strain = 0.08. Other researchers pres-
ent a hyper-elastic material model with E increasing
gradually with strain, yet still the values of initial E
vary considerably. For example, Bryant and McDon-
nell3 have an initial value of 0.08 N/mm2, Zeng et al.36
have E = 0.27 N/mm2 and Nash et al .26 have
E = 20.1 N/mm2 (all at strain = 0.01).
Another likely reason for the variation is the test
method. The strip testing adopted by Nash et al.26 has
a number of inherent geometric inefficiencies related to
the initial curved form of the corneal specimen, which
does not lend itself to strip testing.11 On the other
hand, inflation testing used for example by Bryant and
McDonnell,3 which maintains the cornea in its natural
working condition, is considered more accurate and its
load application speed more representative of the
normal state. For this reason, the test method is
expected to have a significant effect on the values of E
obtained experimentally.
Another factor that adds to the variation in material
models is the highly visco-elastic behaviour of corneal
tissue. This behaviour makes the material properties
dependent on the strain rate used in testing. For this
reason, it is significant that the studies discussed above
adopted different strain rates and in some cases, thestrain rates were not reported.
In this work, a stressstrain relationship of the form
shown in Fig. 3 with an initial E = 0.30 N/mm2 is
used. This relationship has been obtained from a lim-
ited experimental study conducted earlier by the
authors11 and is quite similar to the results reported by
Hoeltzel et al.,21 Kampeier et al.22 and Zeng et al.36
However, as will be seen below, the parametric studies
on effect of CCT and R have been repeated considering
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another material model with initial E = 0.08 N/mm2
to demonstrate the effect of variation in material
properties on GAT IOP. This additional model, which
was obtained from work by Bryant and McDonnell,3 is
also shown in Fig. 3.Note should be made that the material properties
obtained from inflation tests and reported in the lit-
erature were recorded after loading the cornea to the
inflation point. Before this point, the corneal behav-
iour was unstable and the readings remained incon-
sistent until the cornea had taken its natural inflated
form. For this reason and since the numerical models
described in this paper are based on these material
properties, these models can only describe the behav-
iour beyond the inflation point.
RESULTS
Analysis of a Case with Calibration Dimensions
GAT is based on the modified ImbertFick law,
which states that the tonometric pressure, and the effect
of surface tension along the edge of the tonometer
caused by the tear film, equal the IOP and the effect of
the bending resistance of the cornea, see Fig. 4.
Using experimental observations made by Gold-
mann and Schmidt,17 Ehlers et al.9 and Whitacre
et al.,34 it is found that for certain corneal dimensions,
the effect of surface tension cancels out the effect of
bending resistance. As a result, the external tonometry
pressure at applanation, (also called the intra-ocular
pressure measured using Goldmann tonometry, or
IOPG) equals the true intra-ocular pressure, denoted
IOPT. In this special case, the correction factor,
K = IOPG/IOPT equals 1. The corneal dimensions at
which this special case arises are called the calibration
dimensions, namely CCT = 0.520 mm and
R = 7.80 mm. Numerical modelling of GAT starts
with this special case and the value of the numerical
correction factor is compared with the expected value
of 1 found experimentally.
In this work, a corneal model with 17,424 six-noded
solid elements arranged in 6 layers and 22 rings is used.
This large number of elements has been necessary to
model the concentrated effect of tonometry and to
create a fine mesh at the contact area with the
tonometer. The tonometer model, which has a
3.06 mm diameter, uses similar solid elements. The
anterior surface of the cornea and the posterior surface
of the tonometer are described in the analysis as con-
tact surfaces to prevent over-closure of the gap
between them. The edge supports of the cornea are
roller supports set at 23 to the limbal plane in order to
represent the effect of connection with the sclera.10
The material properties used in the analysis have an
initial Youngs modulus, E = 0.30 N/mm2. When
analysed using the Ogden strain formula given in Eq.
(1) and assuming a forth order (N = 4) for improved
accuracy, the following values of u and a parameters
are obtained: u1 =)
110.3, u2 = 55.64, u3 = 108.2,u4 = )53.54, a1 = 14.97, a2 = 16.06, a3 = 12.93,
a4 = 11.99. These values provide a close fit with the
stressstrain relationship with initial E = 0.30 N/mm2
shown in Fig. 3. Note that since the extrafibrillar
matrix of the cornea is principally water, the corneal
tissue is characterized as a nearly incompressible
material.3,19,22,30,33 In Abaqus,22 materials defined as
incompressible are given an elastic volume ratio, Jel, of
1. Further, when the material response is incompress-
ible, the solution to the analysis problem cannot be
obtained in terms of the displacement history only
since a hydrostatic pressure can be added withoutchanging the displacements. This difficulty is removed
in Abaqus by treating the pressure as an independently
interpolated basic solution variable, coupled to the
displacement solution through the constitutive theory
and the compatibility condition, with this coupling
implemented by a Lagrange multiplier. This process is
implemented through the use of hybrid elements that
use a mixture of displacement and stress variables with
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Strain (mm/mm)
Stress(N/mm
2)
Model with initial E = 0.30 N/mm2
Model with initial E = 0.08 N/mm2
FIGURE 3. Material models considered in parametricstudies.
Intr
a-o
cula
rpres
sure
Tonometric pressure
Surface tensionBending resistance
FIGURE 4. Corneal deformation under IOP and tonometricpressure.
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an augmented variational principle to approximate the
equilibrium equations and compatibility conditions.
Further details of this process can be found in the
program documentation.20
The analysis starts by subjecting the corneal model
to an IOP with a predetermined value of 0.002 N/mm2
(15 mmHgapproximately at middle of normal IOP
range), acting as uniform pressure on the internal faces
of the internal layer of solid elements. After inflation
under full IOP, the model is subjected to contact
pressure from the tonometer, which is pushed gradu-
ally and concentrically against the cornea until com-
plete applanation is achieved. The stress distributions
recorded during this process are shown in Fig. 5. It can
be seen that increases in stress are limited to the
tonometer contact area with limited effect elsewhere.
The closure of gap between the tonometer and the
corneal anterior surface is continuously monitored
during the analysis to determine the point at which
applanation has occurred. Figure 6 shows the progress
of gap closure over five stages of the analysis, the lastof which represents the point of full applanation.
The contact stress distribution between the tonom-
eter and the corneal anterior surface is also monitored
during the progress of applanation. It is interesting to
observe how the maximum values of contact stress
change location with the progress of applanation, see
Fig. 7. At the start, the stress is highest at the centre of
the tonometer as this is where the contact is initiated.
Then as applanation progresses, the area of highest
contact stress shifts away from the centre and finally
locates at approximately half the tonometer radius at
full applanation.
At applanation, the force required to push the
tonometer model to this point is divided by the contact
area to obtain the external pressure. This pressure is
produced in actual tonometry by two effects, one is the
tonometry pressure (referred to as Goldmann IOP or
IOPG) and the other is the effect of tear film surface
tension. The value of surface tension is taken as
0.0455 N/m from work carried out at the University of
New South Wales and not yet published. This value is
slightly less than that for water, 0.0728 N/m.
The surface tension acts along the edge of the con-
tact area. Therefore, its effect on the IOPG calculations
is determined by multiplying the surface tension by the
tonometer perimeter (2 1.53p) and dividing it by thecontact area (1.532p). The resulting pressure is
subtracted from the previously calculated externalpressure to calculate IOPG. In the analysis, the true
intra-ocular pressure (IOPT) applied is 0.002 N/mm2
(15 mmHg) and the external pressure is determined as
0.002074 N/mm2. The small effect of tear film surface
tension is 0.0455 10)3 (2 1.53p)/(1.532p) =0.000059 N/mm2. Therefore IOPG = 0.002074
0.000059 = 0.002015 N/mm2 = 15.11 mmHg. This
FIGURE 5. Stress distribution during the analysis: (a) following application of IOP = 0.002 N/mm2 (15 mmHg), (b) following fullapplanationview without tonometer, (c) following full applanationview with tonometer. Contours are drawn on model withoriginal undeformed geometry. Stress range: red= 0.09 N/mm2, blue= 0.00 N/mm2.
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gives a correction factor = 15.11/15 = 1.007. This
value is close to 1.0, which is expected with the cali-
bration dimensions.
Effect of Material Modelling Approaches
The analysis of the above case with the calibration
dimensions has been repeated with the adoption of a
constant E equal to the initial E of the model ( E =
0.30 N/mm2). In this case, applanation is achieved
under an external pressure of 0.001714 N/mm2. After
removing the effect of surface tension, IOPG is found
as 0.001655 N/mm2 or 12.42 mmHgwith an error of
2.58 mmHg or 17.2%. Further study of the behaviour
of the numerical model with both the linear and non-
linear material definitions reveals significant differ-
ences in behaviour. For instance Fig. 8 shows the
anterior displacement of points along a corneal
meridian line starting from the centre. The model with
hyper-elastic material definition demonstrates a non-
linear displacement rate during both inflation and
applanation, giving a strong indication that the stress
regime within the model rose beyond the initial low
stiffness phase of the material. This behaviour is not
detected with linear material definition where the
displacement rate is almost linear and as a result,
applanation is achieved at a different pressure level.
It has therefore been concluded that the nonlinear
material model should be used in all further GAT
modelling since the stresses generated during the
procedure could exceed those associated with the first
low-stiffness phase of material behaviour.
Parametric Study 1Effect of Variation in CCT
Figure 9 shows numerical estimation of the effect of
CCT variation on IOP measurements using GAT
(IOPG). The figure shows the results obtained with
FIGURE 6. Closure of gap between the tonometer and the corneal anterior surface over five stages of analysisthe last stagemarks the full applanation pointFigure shows the tonometer and part of the corneal anterior surface with contours drawn onmodel with original undeformed geometry. Gap width range: red= 0.150 mm, blue= 0.00 mm.
FIGURE 7. Distribution of contact pressure on the tonometer surface with the progress of applanationfigure shows thetonometer and part of the corneal anterior surface with contours drawn on model with original undeformed geometry. Contactstress range: red= 0.0045 N/mm2, blue= 0.00 N/mm2.
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two material models, with initial Youngs modulus,
E = 0.30 and 0.08 N/mm2, respectively. Within the
CCT range considered (between 0.320 and 0.720 mm),
IOPG changes by 7.03 and 2.35 mmHg (or 1.75 and
0.60 mmHg per 0.100 mm variation in CCT) for the
two material models, respectively. In all cases, R is
kept constant at 7.8 mm. The effect of CCT on IOPGfor the first material model (0.0175 mmHg per 1 lm) is
comparable to values obtained earlier in population
based studies. Earlier values include 0.015 and 0.018 by
Foster et al.,15 0.018 and 0.024 by Foster et al.14 and
0.019 mmHg by Wolfs et al.35
The results indicate that IOP is overestimated with
CCT values larger than 0.520 mm and underesti-
mated with CCT values below it. There is also
evidence that the change in material stiffness has a
clear effect on the response to CCT variation. With astiffer material, the corneal resistance increases and
makes the CCT thickness effect on IOPG more
pronounced.
Parametric Study 2Effect of Variation in R
The numerical estimation of the effect of corneal
curvature, as described by the anterior radius, R, on
IOPG is depicted in Fig. 10. In this numerical study,
R is varied between 7.2 and 8.4 mm, while CCT is
kept unchanged at 0.520 mm. The study has been
conducted twice for material models with initial
Youngs modulus of 0.30 and 0.08 N/mm2.
As R increases, the corneal curvature decreases
leading to a reduction in the structural resistance and
hence an underestimation of IOP. With R increasing
from 7.2 to 8.4 mm, IOPG reduces by 1.63 and
1.58 mmHg (or 1.35 and 1.32 mmHg per 1.0 mm
variation in R) for the two material models, respec-
tively. The changes in IOPG are more pronounced
with R below the calibration value of 7.8 mm.
Beforeapplication
ofIOP
Afterapplication
ofIOP
Pointofapplanation
IOP = 0 to 15 mmHg Tonometric pressure
(exceeding point of applanation)
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
Centre point
Intermediate point
Intermediate point
Intermediate point
Intermediate point
Point under edge of tonometer
Point outside tonometer area
Centre point
Intermediate point
Intermediate point
Intermediate point
Intermediate point
Point under edge of tonometer
Point outside tonometer areaAnteriordisplacementofcornealsurfac
e(mm)
Beforeapplication
ofIOP
Afterapplication
ofIOP
Pointofapplanation
IOP = 0 to 15 mmHg Tonometric pressure
(exceeding point of applanation)
7.6
7.8
8.0
8.2
8.4
8.6
8.8
Anteriordisplacementofcornealsurfac
e(mm)
(a) (b)
FIGURE 8. Anterior displacement of the corneal surface under the tonometer during both inflation and applanation. (a) Modelwith nonlinear material properties, (b) model with linear material properties.
12
13
14
15
16
17
18
19
20
21
22
0.3 0.4 0.5 0.6 0.7 0.8
CCT (mm)
IOPG(m
mHg
)
Material model with initial E = 0.30 N/mm2
Material model with initial E = 0.08 N/mm2
FIGURE 9. Numerical estimation of the influence of CCT onIOP measurement using GAT for two material models IOPT = 15 mmHg in all cases.
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Parametric Study 3Effect of Variation in Material
Constitutive Relationship
The third parametric study considers five materialmodels including the models with initial Youngs
modulus, E = 0.30 and 0.08 N/mm2. The other three
models have an initial E = 0.15, 0.60 and 1.20 N/
mm2. The results of considering different material
models on IOPG are illustrated in Fig. 11. The change
in IOPG caused by the maximum change in material
properties (from E = 0.08 to 1.20 N/mm2) is
17.0 mmHg, most of which is observed when the
material model is stiffer than 0.30 N/mm2.
Clinical Data: Methods
The results from the modelling study were com-
pared to those obtained from a clinical dataset from
the Glaucoma Research Unit, Moorfields Eye Hospi-
tal, London. GAT IOP, ultrasound measured CCT
and central corneal curvature measurements were ac-
quired from 532 eyes of 532 new referrals to the Unit
over a 14-month period. None of the patients were
taking topical IOP-lowering medication. Measurement
of CCT and corneal curvature were obtained by one of
four technicians, and GAT IOP measurements were
made by a single experienced clinician. CCT mea-
surements were performed using a contact ultrasound
pachymeter (20 MHz solid tip probe; Optikron 2000,
Roma, Italy). R measurements were made with a
noncontact keratometer (IOLMaster; Carl Zeiss
Meditec, AG, Germany). The device measures the
curvature of a 2.3 mm diameter circular area of the
central cornea. Only one eye per patient was used in
the analysis.
In an attempt to isolate the relative effects of R and
CCT on GAT IOP measurements, two analyses were
performed. The first investigated the effect of varying
CCT on GAT IOP measurements for three R ranges
(110 eyes with R = 6.87.5 mm, 317 eyes with
R = 7.517.9 mm, 105 eyes with R = 7.918.6 mm)
whilst the second assessed the effect ofR on GAT IOP
measurements for three CCT ranges (126 eyes with
CCT = 410530 lm, 290 eyes with CCT = 531
590 lm, 116 eyes with CCT = 591660 lm). The
results are presented graphically in Figs. 12 and 13.
Linear predictive models were chosen to assess the
relationships between corneal parameters and GAT
IOP measurement. The purpose of this exercise was to
assess how clinical findings related to our numerical
studies, and it was felt that a linear analysis would give
a good indication of agreement, if any.
GAT IOP measurements increased with increasing
CCT in all R groups, of the order between 1.6 and
3.3 mmHg per 0.1 mm increase in CCT. The trend was
most significant in the 7.58.6, or flat and middle
ranges of corneal curvature, where CCT accounted for
approximately 7.5% of the measurement variation.
This suggests that although the effect of CCT has astatistically significant effect on GAT IOP measure-
ments, its effect is small compared with other sources
of measurement variation. Other sources of variation
include the technique of IOP measurement in the
clinical setting (which is subject to both inter- and
intra-observer variation), variation in IOPT, variation
in the effect of CCT at different IOPT, and, poten-
tially, variation in corneal material properties, all of
which will add to the spread of data. This measure-
ment imprecision is a source of noise which may mask
the effect of corneal parameters on IOP measurement.
Therefore, it is possible that the effect of CCT may be agreater predictor of GAT IOP measurement than the
findings suggest. The composite graph showing
the linear trends for all three R ranges suggests that the
GAT IOP measurement is somewhat underestimated
12
13
14
15
16
17
18
7 7.2 7.4 7.6 7.8 8 8.2 8.4 8.6
Anterior radius, R (mm)
IOPG(mmHg)
Material model with initial E = 0.30 N/mm2
Material model with initial E = 0.08 N/mm2
FIGURE 10. Numerical estimation of the influence of R onIOP measurement using GAT for two material models.IOPT = 15 mmHg in all cases.
10
15
20
25
30
35
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Initial E (N/mm2)
IOPG(mm
Hg
)
FIGURE 11. Numerical estimation of the influence of mate-rial properties on IOP measurement using GAT.IOPT = 15 mmHg in all cases.
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in flat corneas (R range 7.58.6 mm) compared with
more curved corneas (R range 6.87.5 mm).
Changes in R produce similar effects, but, as in the
numerical model, are not as significant as changes in
CCT, and only account for up to 2% of the measure-
ment error. As R becomes progressively greater, the
result is an underestimation of IOP of the order of
2 mmHg per 1 mm change in R, although the effect is
considerably smaller in the middle range of CCT
between 531 and 590 lm.
Overall, the clinical data suggests that in eyes with
steep (R between 6.8 and 7.5 mm) and thick (591
660 lm) corneas, the GAT IOP measurement is over-
estimated.
DISCUSSION
There is strong evidence that nonlinear material
properties should be adopted in reproducing the behav-
iour of the human cornea under Goldmann tonometry.
The displacement change during both inflation and
applanation is nonlinear, giving an indication that the
stress regime within the model rose beyond the initial low
stiffness phase of the material. Simplifying the analysis
by considering only the initial Youngs modulus leads to
considerable results variation and significant underesti-
mation of intra-ocular pressure.
The numerical model has been successful in
obtaining a correction factor close to 1 for an eye with
the calibration dimensions (CCT = 0.520 mm,
R = 7.8 mm) and with a nonlinear material model
with an initial Eof 0.30 N/mm2. This case was studied
in detail to show that the GAT procedure did not lead
to notable changes in the stress distribution in the
cornea outside the applanated area. The analysis also
showed that the contact pressure between the tonom-
eter and the anterior cornea was not uniform, but
reached its maximum value along a ring with about
half the tonometer radius.
The model with the nonlinear material properties
predicts a clear effect of CCT and R variation on IOP as
measured using Goldmann tonometry. The effect is more
pronounced with CCT values above 0.520 mm and R
IOPG = 0.0162xCCT + 8.8808
R2 = 0.0246
0
5
10
15
20
25
30
35
400 500 600 700 400 500 600 700
400 500 600 700 400 500 600 700
CCT (m)
IOPG(mm
Hg
)
IOPG = 0.0334xCCT - 1.0115
R2 = 0.0746
0
5
10
15
20
25
30
35
CCT (m)
IOPG(mm
Hg
)
(a) (b)
IOPG = 0.0305xCCT + 0.3053
R2 = 0.0761
0
5
10
15
20
25
30
35
CCT (m)
IOPG(
mm
Hg
)
0
5
10
15
20
25
CCT (m)
IOPG(mm
Hg
)
R=6.8 to 7.50R=7.51 TO 7.9
R=7.91 to 8.6
(c) (d)
FIGURE 12. Analysis of clinical observation to determine the effect of CCT variations on intra-ocular pressure readings usingGAT. (a) Observations for patients with anterior radius between 6.8 and 7.5 mm, (b) observations for patients with anterior radiusbetween 7.51 and 7.9 mm, (c) observations for patients with anterior radius between 7.91 and 8.6 mm, (d) linear trend lines for theabove three cases.
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below 7.8 mmcollectively termed the calibration
dimensions. With IOPT of 15 mmHg, a nonlinearmaterial model with initial Youngs modulus, E =
0.30 N/mm2, increasing CCT from 0.520 mm to 0.620
and 0.720 mm (i.e. increases by 19% and 38%) results in
IOP overestimations by 12.0% (16.96)15.11 =
1.85 mmHg) and 30.8% (19.84) 15.11 = 4.73 mmHg),
respectively. Reducing CCT by the same percentages
leads to smaller IOP underestimations of 9.0% (15.11)
13.73 = 1.38 mmHg) and 15.0% (15.11) 12.71 =
2.30 mmHg). The study with a less stiff material model
(initial E = 0.08 N/mm2) shows consistently smaller
effect of CCT variation on IOP measurement.
The effect of changes in R is less notable. Reducing R
from 7.8 mm to 7.5 and 7.2 mm results in IOP overes-
timations of 3.1% (15.59) 15.11 = 0.48 mmHg) and
6.4% (16.07)15.11 = 0.96 mmHg), respectively. On
the other hand, increasing R to 8.1 and 8.4 mm leads to
slightly smaller effects of 3.1% (15.11)14.63 =
0.48 mmHg) and 4.1% (15.11) 14.49 = 0.62 mmHg).
The effects on IOP measurement are again slightly
reduced when using the less stiff material model.
These results are compatible with the previously
reported effect of CCT and R on the structural resistance
of the cornea against applanation. Also as CCT is the
factor with the higher effect on the structural resistance,it is expected that its variation will lead to larger
discrepancies in IOP measurement.
The material model is also found to have a pro-
found effect on IOP measurements when studied sep-
arately. The range of variation in initial Youngs
modulus is varied within a wide range (0.081.2 N/
mm2). The effect of increasing the material stiffness by
a factor of 4 (from 0.30 to 1.2 N/mm2) is found to lead
to a large overestimation of IOP by 100.9%
(30.35) 15.11 = 15.24 mmHg) while reducing the
stiffness by the same factor (from 0.30 to 0.08 N/mm2)
only results in 11.9% underestimation of IOP
(15.11)
13.31 = 1.80 mmHg). It should be noted here
that clinical research is needed to determine the actual
range of variation in material stiffness to be expected
between different people, during the life span of the
same person and even during the different hours of
day.
The results of the numerical study have been com-
pared with the statistical results of the clinical data and
also the earlier mathematical predictions made by Liu
and Roberts25 and by Orssengo and Pye.29 The data
IOPG = -2.335xR + 34.102
R2 = 0.0192
0
5
10
15
20
25
30
35
6.6 7.1 7.6 8.1 8.6 6.6 7.1 7.6 8.1 8.6
6.6 7.1 7.6 8.1 8.6 6.6 7.1 7.6 8.1 8.6
R (mm)
IOPG = -0.2557xR + 19.718
R2 = 0.0002
0
5
10
15
20
25
30
35
R (mm)
IOPG(mm
Hg
)
IOPG
(mm
Hg
)
IOPG
(mm
Hg
)
IOPG(mm
Hg
)
(a) (b)
IOPG = -2.0711xR + 35.302
R2 = 0.0161
0
5
10
15
20
25
30
35
R (mm)
0
5
10
15
20
25
R (mm)
CCT = 410 - 530CCT = 531 - 590
CCT = 591 - 659
(c) (d)
FIGURE 13. Analysis of clinical observation to determine the effect of anterior radius variations on intra-ocular pressure readingsusing GAT. (a) Observations for patients with CCT between 410 and 530 lm, (b) observations for patients with CCT between 531and 590 lm, (c) observations for patients with CCT between 591 and 660 lm, (d) linear trend lines for the above three cases.
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available for comparison were provided by Liu and
Roberts who used CCT = 0.526 mm, R = 7.8 mm
and E = 0.19 N/mm2. Orssengo and Pye used
CCT = 0.520 mm, R = 7.8 mm and Ewas taken as a
function in IOP in the form E = 0.0229 IOPT. The
comparisons are shown in Figs. 1416. The overall
trends in all cases are similar and there is a notable
measure of agreement between the results, particularly
between our data and those obtained by Liu and
Roberts. The clinical data also shows a similar trend to
the numerical results in spite of three possible sources
of error. Firstly, referrals to the Glaucoma Unit are
usually on the basis of high IOP, unlike the numerical
models which have a moderate IOP of 15 mmHg.
Secondly, the clinical dataset is likely to have a higherthan usual percentage of individuals with stiffer and/or
thicker corneas, as these properties result in overesti-
mated IOP measurements. Thirdly, the assumption of
an almost spherical topography in the numerical
models represents an approximation of the actual
corneal topography. However, the match between the
numerical and clinical data remains encouraging and
should be indicative of the validity of the simulations
used.
REFERENCES
1Anderson, K., A. Elsheikh, and T. Newson. Application ofstructural analysis to the mechanical behaviour of thecornea. J. Roy. Soc. Interface 1:113, 2004.2Bennett, G. and R. B. Rabbetts. Clinical Visual Optics,
vol. 2. Butterworths, LondonBennett G., R. B. Rabbetts(1989) Clinical Visual Optics, vol. 2 Butterworths, London.3Bryant, M. R. and P. J. McDonnell. Constitutive laws for
bio-mechanical modelling of refractive surgery. J. Bio-Mech. Eng. 118:473481, 1996.4Cho, P. and S. W. Cheung. Central and peripheral corneal
thickness measured with the TOPCON specular micro-scope SP-2000P. Curr. Eye Res. 21(4):799807, 2000.5Crisfield, M. A.. Non-linear Finite Element Analysis of
Solids and Structures. Wiley, ChichesterCrisfield M. A.(1997) Non-linear Finite Element Analysis of Solids andStructures. Wiley, Chichester.6Doughty, M. J. and M. L Zaman. Human corneal thick-
ness and its impact on intraocular pressure measures: areview and meta-analysis approach. Surv. Ophthalmol.44(5):367408, 2000.7Doughty, M. J., M. Laiquzzaman, A. Mu ller, E. Oblak,
and N. F. Button. Central corneal thickness in European(white) individuals, especially children and the elderly, andassessment of its possible importance in clinical measures
5
7
9
11
13
15
17
19
21
23
25
0.4 0.45 0.5 0.55 0.6 0.65 0.7
CCT (mm)
IOPG(mmHg)
Present workOrssengo and PyeLiu and RobertsClinical data
FIGURE 14. Effect of CCT variation on IOP measurement aspredicted by the present numerical study, the mathematicalmodelling carried out by Liu and Roberts and by Orssengoand Pye, and the statistical analysis of clinical data for eyeswith R between 7.51 and 7.9 mm. In their analysis, Liu andRoberts used R= 7.8 mm and E= 0.19 N/mm2. Orssengo and
Pye used R= 7.8 mm and E = 0.0229 IOPT.
10
15
20
25
30
35
40
0.1 0.3 0.5 0.7 0.9 1.1 1.3
Initial E (N/mm2)
IOPG(mmHg)
Present work
Orssengo and Pye
Liu and Roberts
FIGURE 16. Effect of material stiffness on IOP measurementas predicted by the present numerical study, and the mathe-matical modelling carried out by Liu and Roberts and Ors-sengo and Pye. In their analysis, Liu and Roberts usedCCT = 0.526 mm and R= 7.8 mm. Orssengo and Pye usedCCT = 0.520 mm and R= 7.8 mm. These values were alsoadopted in present numerical analysis.
10
1112
13
14
15
16
17
18
19
20
7 7.5 8.58 9
R (mm)
IOPG(mm
Hg
)
Present workOrssengo and Pye
Liu and RobertsClinical data
FIGURE 15. Effect of R variation on IOP measurement aspredicted by the present numerical study, the mathematicalmodelling carried out by Liu and Roberts and Orssengo andPye, and the statistical analysis of clinical data for eyes withCCT between 460 and 530 lm. In their analysis, Liu andRoberts used CCT = 0.526 mm and E = 0.19 N/mm2. Orssengoand Pye used CCT = 0.520 mm and E= 0.0229 IOPT.
Assessment of Goldmann Tonometry Using Numerical Modelling 1639
-
8/3/2019 Assessment of Goldmann Tonometry Using Numerical Modelling
13/13
of intra-ocular pressure. Ophthal. Physiol. 22:491504,2002.8Dubbelman, M., H. A. Weeber, R. G. L. Van Der Heijde,
and H. J. Vo lker-Dieben. Radius and asphericity of theposterior corneal surface determined by corrected Sche-impflug photography. Acta Ophthalmol. Scand. 80:379383, 2002.9Ehlers, N., T. Bramsen, and S. Sperling. Applanation to-
nometry and central corneal thickness. Acta Ophthalmol.
(Copenh) 53:3443, 1975.10Elsheikh, A. and D. Wang. Numerical modelling of cornealbiomechanical behaviour. Comput. Methods Biomech.Biomed. Eng., in press.
11Elsheikh, A. and K. Anderson. Comparative study ofcorneal strip extensometry and inflation tests. J. Roy. Soc.Interface 2:177185, 2005.
12Feltgen, N., D. Leifert, and J. Funk. Correlation betweencentral corneal thickness, applanation tonometry and di-rect intracameral IOP readings. Br. J. Ophthalmol. 85:8587, 2001.
13Foster, A. and G. J. Johnson. Magnitude and causes ofblindness in the developing world. Int. Ophthalmol.14(3):135140, 1990.
14Foster, P. J., J. Baasanhu, P. H. Alsbirk, D. Munkhbayar,
D. Uranchimeg, and G. J. Johnson. Central cornealthickness and intraocular pressure in a Mongolian popu-lation. Ophthalmology 105:969973, 1998.
15Foster, P. J., D. Machin, T.-Y. Wong, T.-P. Ng, J. F. Ki-rwan, G. J. Johnson, P. T. Khaw, and S. K. L. Seah.Determinants of intraocular pressure and its associationwith glaucomatous optic neuropathy in Chinese Singapo-reans: the Tanjong Pagar study. Invest. Ophthalmol. VisionSci. 44:38853891, 2003.
16Goldmann, H. and T. Schmidt. Uber Applanationstono-metrie. Ophthalmologica 134:221242, 1957.
17Goldmann, H. and T. H. Schmidt. Weiterer beitrag zurapplanationstonometrie. Ophthalmologica 141:441456,1961.
18Gunvant, P., M. Baskaran, L. Vijaya, I. S. Joseph, R. J.
Watkins, M. Nallapothula, D. C. Broadway, and D. J.OLeary. Effect of corneal parameters on measurementsusing the pulsatile ocular blood flow tonograph andGoldmann applanation tonometer. Br. J. Ophthalmol.88:518522, 2004.
19Hanna, K. D., F. E. Jouve, G. O. Waring, and P. G.Ciarlet. Computer simulation of acute keratotomy forastigmatism. Refract. Corneal Surg. 8:52163, 1992.
20Hibbitt, Karlsson, Sorensen Inc. Abaqus: Standard UsersManual. Detroit, USA, 2001.
21Hoeltzel, D. A., P. Altman, K. Buzard, and K.-I. Choe.Choe Strip extensometry for comparison of the mechanicalresponse of bovine, rabbit and human corneas. Trans.ASME 114:202215, 1992.
22Kampmeier, J., B. Radt, R. Birngruber, and R. Brink-mann. Thermal and biomechanical parameters of porcinecornea. Cornea 19(3):355362, 2000.
23Kanngiesser, H. E., C. Inversini, and V. L. Ducry. Simu-lation of dynamic contour tonometry on a non-linear non-spherical eye model using finite element methods. ARVO2005, Poster #1340.
24Lam, A. K. C. and J. S. Chan. Corneal thickness at dif-ferent reference points from Orbscan II system. Clin. Exp.
Optom. 86(4):230234, 2003.25Liu, J. and C. J. Roberts. Influence of corneal biome-chanical properties on intraocular pressure measurement:quantitative analysis. J. Cataract. Refract. Surg. 31:146155, 2005.
26Nash, I. S., P. R. Greene, and C. S. Foster. Comparison ofmechanical properties of keratoconus and normal corneas.Exp. Eye Res. 35:41342, 1982.
27Nooshin, H. and H. Tomatsuri. Diamatic transformations,International Symposium on Spatial Structures: Heritage,Present and Future, Milan, Italy, 1995, pp. 112.
28Ogden, R. H. Non-linear Elastic Deformations. PrenticeHall, 1984.
29Orssengo, G. J. and D. C. Pye. Determination of the trueintraocular pressure and modulus of elasticity of the Hu-
man cornea in vivo. Bull. Math. Biol. 61:551572, 1999.30Pinsky, P. M., D. van der Heide, and D. Chernyak.
Computational modeling of mechanical anisotropy in thecornea and sclera. J. Cataract. Refract. Surg. 31:136145,2005.
31Riks, E. The application of Newtons method to theproblem of elastic stability. J. Appl. Mech. 39:10601066,1972.
32Tonnu, P.-A., T. Ho, T. Newson, A. Elsheikh, K. Sharma,E. White, C. Bunce, and D. Garway-Heath. The influenceof central corneal thickness and age on intraocular pressuremeasured by pneumotonometry, non-contact tonometry,the Tono-Pen XL and Goldmann applanation tonometry.Br. J. Ophthalmol. 89:851854, 2005.
33Vito, R. P., T. J. Shin, and B. E. McCarey. A mechanical
model of the cornea: The effects of physiological, surgicalfactors on radial keratotomy surgery. Refract. CornealSurg. 5:8288, 1989.
34Whitacre, M. M., R. A. Stein, and K. Hassanein. The effectof corneal thickness on applanation tonometry. Am. J.Ophthalmol. 115:592596, 1993.
35Wolfs, R. C., C. C. Klaver, J. R. Vingerling, D. E. Grob-bee, A. Hofman, and P. T. Jong. Distribution of centralcorneal thickness and its association with intraocularpressure: the Rotterdam Study. Am. J. Ophthalmol.123:767772, 1997.
36Zeng, Y., J. Yang, K. Huang, Z. Lee, and X. Lee. Acomparison of biomechanical properties between humanand porcine cornea. J. Biomech. 34:533537, 2001.
ELSHEIKH et al.1640